= Velocity as dependent variable in the Falkner-Skan equation
{title2=$w=f',\quad w^2w_{ff}+w(w_f)^2+\beta f w w_f+m(1-w^2)=0$}
For the <Falkner-Skan equation> $f^{(3)}+\beta f f''+m(1-(f')^2)=0$, an attached branch has $f'>0$ away from the wall. It is then possible to replace the similarity coordinate by $f$ itself. Define $w(f)=f'$. The <chain rule> gives $f''=w w_f$ and $f^{(3)}=w(w_f^2+w w_{ff})$, proving the displayed second-order equation. The wall and outer conditions become $w(0)=0$ and $w(\infty)=1$ when the stream function is set to zero at the wall.
The reduction is singular at the wall: if $s=f''(0)>0$, then $f\sim s\eta^2/2$ and $w\sim\sqrt{2sf}$. Hence $w_f$ diverges while $w w_f\to s$ remains finite. A finite-slope condition on $w$ at zero would incorrectly exclude the ordinary positive-wall-shear branch. At flow reversal or separation, the invertibility of the coordinate change must be checked again.
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